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The indices of refraction, extinction constants and complex conductivities of the GaN film for frequencies ranging from 0.2 to 2.5 THz are obtained using THz time-domain spectroscopy. The results correspond well with the Kohlrausch stretched exponential model. Using the Kohlrausch model fit not only provides the mobility of the free carriers in the GaN film, but also estimates the relaxation time distribution function and average relaxation time.
©2006 Optical Society of America
1. Introduction
Gallium nitride (GaN) is a direct, wide bandgap semiconductor which has been intensely studied over the last ten years. It has multiple applications in blue ultraviolet light-emitting diodes and laser diodes [1–7]. Researchers have used many techniques to analyze the electrical and optical properties of GaN, such as photoluminescence, Raman spectroscopy, magneto-optical infrared transmission, infrared spectroscopic ellipsometry and many others [8–14]. Another area of interest for study of GaN is high temperature/high power electronics [15]. GaN has many excellent electron transport properties, such as good mobility and high saturation drift velocity [16], thus making it suitable for high-frequency transistors operated in the GHz and THz frequency ranges [17,18]. However, the properties of GaN at such high frequencies are less studied. Zhang et al. [19] reported the complex conductivity and refractive index in the THz frequency range of an unintentionally doped GaN. By applying the simple Drude theory, Zhang showed a good theoretical fit to the experimental results. The density and mobility of the free carriers in GaN were also obtained from the noncontact THz time-domain spectroscopy (THz-TDS) measurements. Nagashima et al. [20] studied the temperature dependence of the mobility and dc conductivity of lightly doped GaN films using THz-TDS. The dc conductivities of the GaN films were obtained by fitting the experimental results to the simple Drude model. Nagashima concluded that the dc conductivities of the GaN films obtained from the THz-TDS show good agreement with those obtained using Hall measurements.
The simple Drude model provides a straightforward method of calculating the THz frequency optical constants from the carrier concentration and dc mobility. The time-domain response function of the simple Drude model is the single sided exponential exp(-t/τ), where τ is the characteristic relaxation time. The simple Drude model assumes that the velocity of the carriers is relaxed with a single, energy-independent relaxation time τ and the carriers scattering is isotropic. The drawback of the simple Drude model is that it cannot provide the momentum relaxation distribution function of the free carriers in semiconductors. In addition, some experimental results have shown deviations from the Drude model [21,22]. Thus, it is worthwhile to employ other models that use a distribution of relaxation times, which may be more reflective of real world conditions. Smith [23] proposed that charge carriers scattering might be anisotropic and found that this effectively leads to a generalized Drude model with a different time constant. Other generalized Drude models have also been discussed by Smith et al. [23]. Futhermore, two empirical fitting models that describe the relaxation behavior in a variety of materials have gained widespread acceptance: the Cole-Davidson (CD) model [24] and the Kohlrausch stretched exponential model [25]. The CD model has been extensively used to describe frequency-domain relaxation data. The Kohlrausch stretched exponential model was first used to describe mechanical relaxation data [25]. Williams and Watts then applied it to dielectric relaxation in polymers [26]. The Kohlrausch stretched exponential model can describe time-domain relaxation data. For doped silicon, Jeon and Grischkowsky [21] reported that only the CD model can fit the measurements over the entire frequency range. For the Kohlrausch stretched exponential model, there have been no past published reports of applications to fit the THz transmission measurements of semiconductors.
We studied the complex conductivity and optical constants from 0.2 to 2.5 THz on n-type GaN thin film using THz-TDS. We applied the Drude model, the CD model and the Kohlrausch stretched exponential model to fit the measurements, with the Kohlrausch model providing the best fit. Using the Kohlrausch model fit not only provides the mobility of the free carriers in the GaN film, but also estimates the relaxation time distribution function and average relaxation time.
2. Theoretical background
2.1. Calculating the optical constants and complex conductivity
Figure 1 shows a simple schematic of the properties of interest. The discussion is limited in the frequency domain. Eo(ω) is the incident THz radiation field, Eref(ω) is the reference field (substrate only), and Efilm(ω) is the signal (with the film) field. d is the thickness of the film. n1, n2 and n3 are the refractive indices. In this study, n1=1 and n3 is the refractive index of sapphire. The complex refractive index of thin film is n=n2+iκ2, where n2 is the real refractive index and κ2 is the extinction coefficient. The reference field can be determined by
By considering the multiple reflection within the film, the Efilm(ω) can be represented by
where c is the speed of light in a vacuum and rij and tij are the reflection and transmission coefficients of the i→j interface (i,j=1, 2, and 3, and i≠j), respectively. For a thin film, i.e. |nωd/c|≪1, we have
Equation (3) is denoted as Efilm(ω)/Eref(ω)≠ρei Δ and ε′+iε″≠(n2+iκ2)2, then the real and imaginary parts of Eq. (3) can be represented by the following equations
If both ρ and Δ are obtained from the Fourier transformed reference and signal THz pulses, then the complex refractive index and conductivity of thin film can be deduced by
Fig. 1. n1, n, and n3 are the refractive indices. E0(ω) is the incident THz field, and Eref(ω) and Efilm(ω) are the reference (substrate only) and signal (with the film) THz fields, respectively. d is the thickness of the film.
2.2. The simple Drude, the CD and the Kohlrausch stretched exponential models
The frequency dependence of the complex dielectric constant ε is equal to the square of the complex refractive index. The dielectric constant of the doped GaN sample is described by
where εGaN is the dielectric constant of undoped GaN, σ is the complex conductivity, and ε0 is the free space permittivity. For this work, the dielectric constant of undoped GaN is 9.4 [19]. The dielectric constant described by Eq. (9) is independent of the function form of the conductivity. The complex conductivity of the generalized Drude (GD) model is given by [23]
where ωp is the plasma frequency, τ0 is a characteristic relaxation time, and the exponents α and γ are parameters ranging between 0 and 1. The plasma frequency ωp is defined by , where N is the electron density, e is the electron charge, m*=0.22me for the effective electron mass in GaN [27], and me is the free-electron mass. If α=1 and γ=1, Eq. (10) reduces to the simple Drude model. For the CD model, set α=1 and vary only γ. These models (GD, CD, and the simple Drude) are functions of frequency.
The CD model has the associated relaxation time distribution function gCD(τ) for τ<τ 0 [24,28],
which is defined to be zero for τ≥τ 0. The (gCD(τ)/τ)dτ is proportional to the probability for the carrier relaxation time to be found in a narrow interval dτ around τ. The average relaxation time 〈τ〉 is given by [28]
On the other hand, the Kohlrausch stretched exponential exp(-(t/τk)β), where τk is a characteristic relaxation time and the exponent β is a parameter ranging between 0 and 1, is a function of time. In order to analyze the Kohlrausch stretched exponential in the frequency domain, a Fourier transform is needed. However, it is well known that the Kohlrausch stretched exponential does not have an analytical function form in the frequency domain. The relationship between the parameters of the Kohlrausch stretched exponential model and the GD model was reported by Alvarez [29], and can be shown as [29,30]
then
and
Thus, the above relationships can be used to fit the data in the frequency domain with the GD model using a single independent parameter α.
The relaxation time distribution function for the Kohlrausch stretched exponential model is given by the following series form: [29]
The (gk(τ)/τ)dτ is proportional to the probability for the carrier relaxation time to be found in a narrow interval dτ around τ. The average relaxation time is given in terms of the Γ function as [29]
3. Experiments
The 4.48 µm-thick, n-type, GaN film used in this study was grown by metalorganic chemical vapor deposition on c-plane sapphire. The sample was wurtzite in structure with its uniaxially optical axis perpendicular to the surface. There is a GaN buffer of about 30 nm between the GaN film and its substrate. Room-temperature photoluminescence was detected under excitation with the 325 nm wavelength of a He-Cd laser. The photoluminescence maximum was at 3.41 eV with a FWHM of 61 meV. Electrical properties were characterized by Hall measurement. Carrier concentration and Hall mobility of the electron were 1.47×1018 cm-3 and 300 cm2/Vs, respectively. All measurements were taken at room temperature.
Figure 2 shows a schematic of the experimental setup for THz-TDS, similar to the conventional THz system. A mode-locked amplified Ti:sapphire laser (λ=800 nm) with 0.6 W average output power generated ~130 fs pulses at a repetition rate of 1 kHz. The output laser beam was divided into two -- a pump and a probe. The pump beam was focused on a ~100 µm diameter area on a 100-µm-thick (110) ZnTe crystal. The generated THz radiation was collimated and focused on the sample by a pair of off-axis parabolic mirrors. The transmitted THz radiation through the sample was collimated and focused again on a detector by a second pair of off-axis parabolic mirrors. A 700-µm-thick (110) ZnTe crystal was used in an electro-optic sampling setup for the detection of the THz radiation. The temporal profile of the THz field was mapped out by varying the delay between the pump pulse and the probe pulse.
4. Results and discussion
In order to obtain the optical constants of the GaN film, we also need the refractive indices of the sapphire substrate. These constants were measured using the same apparatus. We found that when the frequency ranged from 0.2 to 2.5 THz, the real refractive indices of sapphire extended from 3.06 to 3.08. The extinction coefficients of sapphire were very small and could be neglected at these frequencies. Our results are consistent with the values reported by D. Grischkowsky et al. [31].
Figure 3 plots the measured THz pulses transmitted through the GaN sample and the sapphire substrate. The oscillations after the main pulse are due to the resonance of the absorption lines of the water vapor. The echoes indicated by the arrows are due to multiple reflections in the sample. The THz pulse spectrum can be obtained by applying a fast Fourier transform to the time-domain waveform. For simplicity, we did not take into account the multiple reflections in the sample and the oscillations after the main THz pulse. Thus, the extraction was done with the data values recorded only up to ~14 ps.
The measured indices of refraction and extinction constants of the GaN film are shown in Figs. 4(a) and 4(b), respectively. As shown in Figs. 4(a) and 4(b), the optical constants of the GaN film decrease with increasing frequency. The measured real and imaginary conductivities of the GaN film are shown in Figs. 4(c) and 4(d), respectively. The real conductivity decreases with increasing frequency. The imaginary conductivity increases with increasing frequency. Because of the limited THz beam power, the data is reliable only above 0.2 THz and below 2.5 THz. The measured indices of refraction, extinction coefficients, and complex conductivities were fit using the simple Drude model, the CD model and the Kohlrausch stretched exponential model. Clearly the Kohlrausch stretched exponential model gives a best fit to the experimental results.
For the simple Drude model with α=1 and γ=1 in Eq. (10), only one fitting parameter was used: the carrier relaxation time τ0=5.01×1014 s. The mobility µ given by µ=eτ 0/m* was calculated directly from the fitting parameter τ 0. Thus µ=400 cm2/Vs was obtained. The plasma frequency was obtained from the Hall measurements and set at ωp=63.25 THz. There is no the relaxation time distribution function for the simple Drude model. Thus the average relaxation time 〈τ〉 was the same as τ 0 at 5.01×10-14 s.
Fig. 4. (a) Plot of the measured the indices of refraction, on a logarithmic scale, for the GaN film. (b) The measured the extinction coefficients, on a logarithmic scale, for GaN film. (c) The measured the real conductivities, on a logarithmic scale, for the GaN film. (d) The measured the imaginary conductivities, on a logarithmic scale, for the GaN film. (--- dash) Fit to the simple Drude model; (... dot) Fit to the Cole-Davidson model; (- solid) Fit to the stretched exponential model.
For the CD model with α=1 in Eq. (10), two fitting parameters were used: the carrier relaxation time τ 0=5.05×10-14 s and the exponent γ=0.79. The plasma frequency was set at ωp=63.25 THz, again obtained from the Hall measurements. There is a relaxation time distribution function for the CD model. According to Eq. (12), the average relaxation time 〈τ⃗ was 3.99×10-14 s. If the characteristic mobility is calculated by µ=e〈τ〉/m*, then the value µ=319 cm2/Vs is obtained.
For the GD model, two fitting parameters were used: the carrier relaxation time τ 0=4.90×10-14 s and the exponent α=0.83. Thus, γ=0.59 can be calculated from Eq. (3). The plasma frequency was also chosen to be ωp=63.25 THz. The exponent β and the relaxation time τk for the Kohlrausch stretched exponential model were calculated from Eqs. (14) and (15), respectively. Thus β was 0.56 and τk was 3.56×10-14 s. According to Eq. (17), the average relaxation time 〈τ〉 was 5.88×10-14 s. Again, if the characteristic mobility is calculated by µ=e〈τ〉/m*, then the value µ=470 cm2/Vs is obtained. All of the fitting parameters are shown in Table 1, along with the corresponding carrier densities, mobilities and average relaxation times. Comparison to the Hall measurements shows a discrepancy between the characteristic mobility obtained here and Hall mobility. It is important to note that the characteristic mobility does not contain the Hall scattering factor and thus deviates from the Hall mobility
The simple Drude, the CD and the Kohlrausch stretched exponential models were applied to fit the measurements over the entire frequency range. We found that the differences between the stretched exponential model and experimental results are smallest.
Table 1. The fitting parameters for the measurements plotted in Fig. 4 and the results for Hall measurements.
According to Eq. (16), the distribution functions of the carrier relaxation time for the Kohlrausch stretched exponential model are presented in Fig. 5. It is apparent that the Kohlrausch stretched exponential distribution function has a longer time extension. The maximum value of the distribution function was normalized to 1 and occurred under the condition that τ 0=τmax. The value of τmax for the stretched exponential model is 2.97×10-14 s. The distribution function of the carrier relaxation time is correlated with the carrier scattering mechanisms. A more rigorous analysis for those relationships will be undertaken in future work.
Fig. 5. Plot of the stretched exponential distribution function against log10(τ/τk) using β=0.56 and τk=3.56×10-14 s.
5. Summary
In conclusion, the indices of refraction, extinction constants and complex conductivities of the GaN film for frequencies ranging from 0.2 to 2.5 THz obtained using THz time-domain spectroscopy are a good fit with the Kohlrausch stretched exponential model. The Kohlrausch stretched exponential model fit then not only provides the mobility of the free carriers in the GaN film, but also estimates the relaxation time distribution function and average relaxation time.
Acknowledgments
This work was supported in part by the National Science Council of R.O.C. under grants No. NSC93-2120-M-019-002 and NSC94-2120-M-019-003 and by the National Taiwan Ocean University under grants No. NTOU-AF94-05-04-09-01 and NTOU-AF94-05-04-01-03.
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